P. Nestler, N. Schlömer, O. Klein, J. Sprekels, F. Tröltzsch, Optimal control of semiconductor melts by traveling magnetic fields, Vietnam Journal of Mathematics, 47 (2019), pp. 793--812, DOI 10.1007/s10013-019-00355-5 .AbstractIn this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new industrial heater-magnet module, the Lorentz forces have a stronger impact on the melt than in earlier technologies. It is known from experiments that during the growth process temperature oscillations with respect to time occur in the neighborhood of the solid-liquid interface. These oscillations may strongly influence the quality of the growing single crystal. As it seems to be impossible to suppress them completely, the main goal of optimization has to be less ambitious, namely, one tries to achieve oscillations that have a small amplitude and a frequency which is sufficiently high such that the solid-liquid interface does not have enough time to react to the oscillations. In our approach, we control the oscillations at a finite number of selected points in the neighborhood of the solidification front. The system dynamics is modeled by a coupled system of partial differential equations that account for instationary heat condution, turbulent melt flow, and magnetic field. We report on numerical methods for solving this system and for the optimization of the whole process. Different objective functionals are tested to reach the goal of optimization.

D. Hömberg, K. Krumbiegel, N. Togobytska, Optimal control of multiphase steel production, Journal of Mathematics in Industry, 9 (2019), pp. 1--32, DOI 10.1186/s13362-019-0063-x .AbstractAn optimal control problem for the production of multiphase steel is investigated, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the ferrite phase fraction. The optimal control problem is analyzed and the first-order necessary and second-order sufficient optimality conditions are derived. For the numerical solution of the control problem reduced sequential quadratic programming (rSQP) method with a primal-dual active set strategy (PDAS) was applied. The numerical results were presented for the optimal control of a cooling line for production of hot rolled Mo-Mn dual phase steel.

L. Adam, M. Hintermüller, Th.M. Surowiec, A PDE-constrained optimization approach for topology optimization of strained photonic devices, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 19 (2018), pp. 521--557, DOI 10.1007/s11081-018-9394-5 .AbstractRecent studies have demonstrated the potential of using tensile-strained, doped Germanium as a means of developing an integrated light source for (amongst other things) future microprocessors. In this work, a multi-material phase-field approach to determine the optimal material configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an “optimal" configuration is one in which the strain in a predetermined minimal optical cavity within the Germanium is maximized according to an appropriately chosen objective functional. Due to manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a socalled aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation (PDE) and manufacturing constraints. The resulting problem is analyzed and solved numerically. The theory portion includes a proof of existence of an optimal topology, differential sensitivity analysis of the displacement with respect to the topology, and the derivation of first and second-order optimality conditions. For the numerical experiments, an array of first and second-order solution algorithms in function-space are adapted to the current setting, tested, and compared. The numerical examples yield designs for which a significant increase in strain (as compared to an intuitive empirical design) is observed.

K. Sturm, M. Hintermüller, D. Hömberg, Distortion compensation as a shape optimisation problem for a sharp interface model, Computational Optimization and Applications. An International Journal, 64 (2016), pp. 557--588.AbstractWe study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.

D. Hömberg, Q. Liu, J. Montalvo-Urquizo, D. Nadolski, Th. Petzold, A. Schmidt, A. Schulz, Simulation of multi-frequency-induction-hardening including phase transitions and mechanical effects, Finite Elements in Analysis and Design, 121 (2016), pp. 86--100.AbstractInduction hardening is a well known method for the heat treatment of steel components. With the concept of multi-frequency hardening, where currents with two different frequency components are provided on a single inductor coil, it is possible to optimize the hardening zone to follow a given contour, e.g. of a gear. In this article, we consider the simulation of multi-frequency induction hardening in 3D. The equations to solve are the vector potential formulation of Maxwell's equations describing the electromagnetic fields, the balance of momentum to determine internal stresses and deformations arising from thermoelasticity and transformation induced plasticity, a rate law to determine the distribution of different phases and the heat equation to determine the temperature distribution in the workpiece. The equations are solved using adaptive finite element methods. The simulation results are compared to experiments for discs and for gears. A very good agreement for the hardening profile and the temperature is observed. It is also possible to predict the distribution of residual stresses after the heat treatment.

D. Hömberg, Th. Petzold, E. Rocca, Analysis and simulations of multifrequency induction hardening, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 84--97.AbstractWe study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwell's equations coupled with an internal energy balance and an ODE for the volume fraction of austenite, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system.

W. Bleck, D. Hömberg, U. Prahl, P. Suwanpinij, N. Togobytska, Optimal control of a cooling line for production of hot rolled dual phase steel, Steel Research International, 85 (2014), pp. 1328--1333.AbstractIn this article, the optimal control of a cooling line for production of dual phase steel in a hot rolling process is discussed. In order to achieve a desired dual phase steel microstructure an optimal cooling strategy has to be found. The cooling strategy should be such that a desired final distribution of ferrite in the steel slab is reached most accurately. This problem has been solved by means of mathematical control theory. The results of the optimal control of the cooling line have been verified in hot rolling experiments at the pilot hot rolling mill at the Institute for Metal Forming (IMF), TU Bergakademie Freiberg.

D. Hömberg, S. Lu, K. Sakamoto, M. Yamamoto, Parameter identification in non-isothermal nucleation and growth processes, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 30 (2014), pp. 035003/1--035003/24.AbstractWe study non-isothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an energy balance to account for recalescence effects. The main novelty of our work is the identification of temperature dependent nucleation rates. We prove that such rates can be uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation.

K. Chełminski, D. Hömberg, O. Rott, On a thermomechanical milling model, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 12 (2011), pp. 615--632.AbstractThis paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions.

A. Fasano, D. Hömberg, D. Naumov, On a mathematical model for laser-induced thermotherapy, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems. Elsevier Science Inc., New York, NY. English, English abstracts., 34 (2010), pp. 3831--3840.AbstractWe study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.

D. Hömberg, Ch. Meyer, J. Rehberg, W. Ring, Optimal control for the thermistor problem, SIAM Journal on Control and Optimization, 48 (2010), pp. 3449--3481.AbstractThis paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.

A. Fasano, D. Hömberg, L. Panizzi, A mathematical model for case hardening of steel, Mathematical Models & Methods in Applied Sciences, 19 (2009), pp. 2101--2126.AbstractA mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions. We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.

D. Hömberg, D. Kern, The heat treatment of steel --- A mathematical control problem, Materialwissenschaft und Werkstofftechnik, 40 (2009), pp. 438--442.AbstractThe goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with machine-based control.

D. Hömberg, N. Togobytska, M. Yamamoto, On the evaluation of dilatometer experiments, Applicable Analysis. An International Journal, 88 (2009), pp. 669--681.AbstractThe goal of this paper is a mathematical investigation of dilatometer experiments to measure the kinetics of solid-solid phase transitions in steel upon cooling from the high temperature phase. Usually, the data are only used for measuring the start and end temperature of the phase transition. In the case of several coexisting product phases, lavish microscopic investigations have to be performed to obtain the resulting fractions of the different phases. In contrast, we show that the complete phase transition kinetics including the final phase fractions are uniquely determined by the dilatometer data and present some numerical identification results.

P. Suwanpinij, N. Togobytska, Ch. Keul, W. Weiss, U. Prahl, D. Hömberg, W. Bleck, Phase transformation modeling and parameter identification from dilatometric investigations, Steel Research International, 79 (2008), pp. 793--799.AbstractThe goal of this paper is to propose a new approach towards the evaluation of dilatometric results, which are often employed to analyse the phase transformation kinetics in steel, especially in terms of continuous cooling transformation (CCT) diagram. A simple task of dilatometry is deriving the start and end temperatures of the phase transformation. It can yield phase transformation kinetics provided that plenty metallographic investigations are performed, whose analysis is complicated especially in case of several coexisting product phases. The new method is based on the numerical solution of a thermomechanical identification problem. It is expected that the phase transformation kinetics can be derived by this approach with less metallographic tasks. The first results are remarkably promising although further investigations are required for the numerical simulations.

Beiträge zu Sammelwerken

M. Hintermüller, N. Strogies, On the consistency of Runge--Kutta methods up to order three applied to the optimal control of scalar conservation laws, in: Numerical Analysis and Optimization, M. Al-Baali, L. Grandinetti, A. Purnama, eds., 235 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2019, pp. 119--154.AbstractHigher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.

O. Rott, P. Rasper, D. Hömberg, E. Uhlmann, A milling model with thermal effects including the dynamics of machine and work piece, B. Denkena, ed., Proceedings, 1st International Conference on Process Machine Interactions, Hannover, September 3--4, 2008, PZH Produktionstechnisches Zentrum GmbH, Garbsen, 2008, pp. 369--378.AbstractThis paper deals with the development of a new mathematical model that characterizes the structure-process interaction for a complex milling system. The structure is divided into a work piece and a machine part, which are represented by different models. While the machine dynamics is characterized by a standard multi-body system, the work piece is described as a linear thermo-elastic continuum. The coupling of both parts is carried out by an empirical process model permitting an estimate of heat and coupling forces occurring during milling. This work reports the derivation of the governing equations emphasizing the coupling and summarizes the numerical algorithms being applied to solve the coupled equation system. The results of numerical simulations that show the dynamics of the complex thermo-mechanical system are presented at the end.

Preprints, Reports, Technical Reports

M.J. ArenasJaén, D. Hömberg, R. Lasarzik, P. Mikkonen, Th. Petzold, Modelling and simulation of flame cutting for steel plates with solid phases and melting, Preprint no. 2670, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2670 .Abstract, PDF (11 MByte)The goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiebaud [1] and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed.

C. Grässle, M. Hintermüller, M. Hinze, T. Keil, Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities, Preprint no. 2617, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2617 .Abstract, PDF (11 MByte)We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

N. Alia, Modeling and optimization of a gas-stirred liquid flow for steelmaking processes, The 20th European Conference on Mathematics for Industry (ECMI), MS27: MSO for steel production and manufacturing, June 18 - 22, 2018, University Budapest, Institute of Mathematics at Eötvös Loránd, Hungary, June 19, 2018.

M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinite-dimensional Systems, March 14 - 18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

D. Hömberg, European collaboration in Industrial and Applied Mathematics, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 38 ``Maths in HORIZON 2020 and Beyond'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Spain, June 15, 2016.

D. Hömberg, Math for steel production and manufacturing, MACSI10 -- Empowering Industrial Mathematical and Statistical Modelling for the Future, December 8 - 9, 2016, University of Limerick, Ireland, December 9, 2016.

CH. Landry, A minimum time control problem for the collision-free robot motion planning, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Optimal Control and Applications, April 18 - 21, 2011, Technische Universität Graz, Austria, April 20, 2011.

CH. Landry, A back-face culling to speed up the resolution of an optimal control problem in robot motion planning, ENUMATH 2011, September 5 - 9, 2011, University of Leicester, UK, September 5, 2011.

O. Rott, Analysis of uncertainties in the stability prediction for milling processes, 2nd International Conference ``Process Machine Interactions'', June 10 - 11, 2010, The University of British Columbia, Vancouver, Canada, June 11, 2010.

D. Hömberg, Coupling of process, machine, and work-piece in production processes --- A challenge for industrial mathematics, Workshop ``Industrial Mathematics and its Practice'', February 23 - 24, 2009, University of Tokyo, Japan, February 23, 2009.

O. Rott, Modeling, analysis and stability of milling processes including workpiece effects, The European Consortium for Mathematics in Industry (ECMI 2008), June 30 - July 4, 2008, University College, London, UK, July 3, 2008.

D. Hömberg, On a mathematical model for high-speed milling including the dynamics of machine and work-piece, Conference ``Direct, Inverse and Control Problems for PDE's'' (DICOP 08), September 22 - 26, 2008, Cortona, Italy, September 26, 2008.

D. Hömberg, Optimal control of semilinear parabolic equations and an application to laser material treatments (part I), University of Tokyo, Department of Mathematical Sciences, Japan, February 21, 2007.

D. Hömberg, Optimal control of semilinear parabolic equations and an application to laser material treatments (part II), University of Tokyo, Department of Mathematical Sciences, Japan, February 22, 2007.